5 problems

Let $c_1,c_2,\ldots,c_k$ be integers, and let $M_q(c_1,c_2,\ldots,c_k)$ be the corresponding matrix over $\mathbb{Z}[q,q^{-1}]$. Its trace is a Laurent polynomial in…

Let $G_s(n,k)$ be the recurrence obtained by summing over partitions of $k$ having size at most $s$, with $q$ an indeterminate and $n,k,s$ in the domain where this recurrence is de…

For $n\geqslant 2$, define … where $[m]!=(1-q)(1-q^2)\cdots(1-q^m)$. Unimodality conjecture. The polynomial $P_n(q)$ is unimodal. The polynomial is known to be reciprocal, and the…

Let $P(x)=a_0+a_1x+\cdots+a_mx^m$ be a polynomial with nonnegative and nondecreasing coefficients. A polynomial is unimodal when its coefficient sequence is unimodal. Alvarez et al…

Let $P(x)=a_0+a_1x+\cdots+a_mx^m$ be a polynomial with nonnegative and nondecreasing coefficients. For $d>0$, let $M_*(P,d)$ and $M^*(P,d)$ denote respectively the smallest and gre…